Schouten–Nijenhuis Bracket

In differential geometry, the Schouten–Nijenhuis bracket, also known as the Schouten bracket, is a type of graded Lie bracket defined on multivector fields on a smooth manifold extending the Lie bracket of vector fields. There are two different versions, both rather confusingly called by the same name. The most common version is defined on alternating multivector fields and makes them into a Gerstenhaber algebra, but there is also another version defined on symmetric multivector fields, which is more or less the same as the Poisson bracket on the cotangent bundle. It was discovered by Jan Arnoldus Schouten (1940, 1953) and its properties were investigated by his student Albert Nijenhuis (1955). It is related to but not the same as the Nijenhuis–Richardson bracket and the Frölicher–Nijenhuis bracket.

Read more about Schouten–Nijenhuis Bracket:  Definition and Properties, Generalizations

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Schouten–Nijenhuis Bracket - Generalizations
... There is a common generalization of the Schouten–Nijenhuis bracket for alternating multivector fields and the Frölicher–Nijenhuis bracket due to ... A version of the Schouten–Nijenhuis bracket can also be defined for symmetric multivector fields in a similar way ... T*(M) of M that are polynomial in the fiber, and under this identification the symmetric Schouten–Nijenhuis bracket corresponds to the Poisson bracket of ...